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# 48. fractals

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## 48.1 Introduction to fractals

This package defines some well known fractals:

• with random IFS (Iterated Function System): the Sierpinsky triangle, a Tree and a Fern
• Complex Fractals: the Mandelbrot and Julia Sets
• the Koch snowflake sets
• Peano maps: the Sierpinski and Hilbert maps

For questions, suggestions and bugs, please feel free to contact me at pepe DOT ramirez AAATTT uca DOT es

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## 48.2 Definitions for IFS fractals

Some fractals can be generated by iterative applications of contractive affine transformations in a random way; see Hoggar S. G., "Mathematics for computer graphics", Cambridge University Press 1994.

We define a list with several contractive affine transformations, and we randomly select the transformation in a recursive way. The probability of the choice of a transformation must be related with the contraction ratio.

You can change the transformations and find another fractal

Function: sierpinskiale (n)

Sierpinski Triangle: 3 contractive maps; .5 contraction constant and translations; all maps have the same contraction ratio. Argument n must be great enougth, 10000 or greater.

Example:

```(%i1) load(fractals)\$
(%i2) n: 10000\$
(%i3) plot2d([discrete,sierpinskiale(n)], [style,dots])\$
```
Function: treefale (n)

3 contractive maps all with the same contraction ratio. Argument n must be great enougth, 10000 or greater.

Example:

```(%i1) load(fractals)\$
(%i2) n: 10000\$
(%i3) plot2d([discrete,treefale(n)], [style,dots])\$
```
Function: fernfale (n)

4 contractive maps, the probability to choice a transformation must be related with the contraction ratio. Argument n must be great enougth, 10000 or greater.

Example:

```(%i1) load(fractals)\$
(%i2) n: 10000\$
(%i3) plot2d([discrete,fernfale(n)], [style,dots])\$
```

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## 48.3 Definitions for complex fractals

Function: mandelbrot_set (x, y)

Mandelbrot set.

Example:

This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.

```(%i1) load(fractals)\$
(%i2) plot3d (mandelbrot_set, [x, -2.5, 1], [y, -1.5, 1.5],
[gnuplot_preamble, "set view map"],
[gnuplot_pm3d, true],
[grid, 150, 150])\$
```
Function: julia_set (x, y)

Julia sets.

This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.

Example:

```(%i1) load(fractals)\$
(%i2) plot3d (julia_set, [x, -2, 1], [y, -1.5, 1.5],
[gnuplot_preamble, "set view map"],
[gnuplot_pm3d, true],
[grid, 150, 150])\$
```

See also `julia_parameter`.

Option variable: julia_parameter

Default value: `%i`

Complex parameter for Julia fractals. Its default value is `%i`; we suggest the values `-.745+%i*.113002`, `-.39054-%i*.58679`, `-.15652+%i*1.03225`, `-.194+%i*.6557` and `.011031-%i*.67037`.

Function: julia_sin (x, y)

While function `julia_set` implements the transformation `julia_parameter+z^2`, function `julia_sin` implements `julia_parameter*sin(z)`. See source code for more details.

This program runs slowly because it calculates a lot of sines.

Example:

This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.

```(%i1) load(fractals)\$
(%i2) julia_parameter:1+.1*%i\$
(%i3) plot3d (julia_sin, [x, -2, 2], [y, -3, 3],
[gnuplot_preamble, "set view map"],
[gnuplot_pm3d, true],
[grid, 150, 150])\$
```

See also `julia_parameter`.

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## 48.4 Definitions for Koch snowflakes

Function: snowmap (ent, nn)

Koch snowflake sets. Function `snowmap` plots the snow Koch map over the vertex of an initial closed polygonal, in the complex plane. Here the orientation of the polygon is important. Argument nn is the number of recursive applications of Koch transformation; nn must be small (5 or 6).

Examples:

```(%i1) load(fractals)\$
(%i2) plot2d([discrete,
snowmap([1,exp(%i*%pi*2/3),exp(-%i*%pi*2/3),1],4)])\$
(%i3) plot2d([discrete,
snowmap([1,exp(-%i*%pi*2/3),exp(%i*%pi*2/3),1],4)])\$
(%i4) plot2d([discrete, snowmap([0,1,1+%i,%i,0],4)])\$
(%i5) plot2d([discrete, snowmap([0,%i,1+%i,1,0],4)])\$
```

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## 48.5 Definitions for Peano maps

Continuous curves that cover an area. Warning: the number of points exponentially grows with n.

Function: hilbertmap (nn)

Hilbert map.

Argument nn must be small (5, for example). Maxima can crash if nn is 7 or greater.

Example:

```(%i1) load(fractals)\$
(%i2) plot2d([discrete,hilbertmap(6)])\$
```
Function: sierpinskimap (nn)

Sierpinski map.

Argument nn must be small (5, for example). Maxima can crash if nn is 7 or greater.

Example:

```(%i1) load(fractals)\$
(%i2) plot2d([discrete,sierpinskimap(6)])\$
```

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